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18
Moduli Spaces of Commutative Ring Spectra
, 2003
"... Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as ..."
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Cited by 19 (0 self)
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Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E#E is flat over E# . We wish to address the following question: given a commutative E# algebra A in E#Ecomodules, is there an E# ring spectrum X with E#X = A as comodule algebras? We will formulate this as a moduli problem, and give a way  suggested by work of Dwyer, Kan, and Stover  of dissecting the resulting moduli space as a tower with layers governed by appropriate AndreQuillen cohomology groups. A special case is A = E#E itself. The final section applies this to discuss the LubinTate or Morava spectra En .
(Pre)sheaves of Ring Spectra over the Moduli Stack of Formal Group Laws
, 2004
"... In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem. ..."
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Cited by 16 (1 self)
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In the first part of this article, I will state a realization problem for diagrams of structured ring spectra, and in the second, I will discuss the moduli space which parametrizes the problem.
topological AndréQuillen homology and stabilization
 Topology Appl. 121 (2002) No.3
"... The quest for an obstruction theory to E ∞ ring structures on a spectrum has led a number of authors to the investigation of homology in the category of E ∞ algebras. In this note we present three, apparently very different, constructions and show that when specialized to commutative rings they all ..."
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Cited by 13 (1 self)
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The quest for an obstruction theory to E ∞ ring structures on a spectrum has led a number of authors to the investigation of homology in the category of E ∞ algebras. In this note we present three, apparently very different, constructions and show that when specialized to commutative rings they all agree. (AMS subject classification 55Nxx. Key words: AndréQuillen homology, E∞homology).
Moduli problems for structured ring spectra
 DANIEL DUGGER AND BROOKE
, 2005
"... In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theore ..."
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Cited by 11 (0 self)
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In this document we make good on all the assertions we made in the previous paper “Moduli spaces of commutative ring spectra ” [20] wherein we laid out a theory a moduli spaces and problems for the existence and uniqueness of E∞ring spectra. In that paper, we discussed the the HopkinsMiller theorem on the LubinTate or Morava spectra En; in particular, we showed how to prove that the moduli space of all E ∞ ring spectra X so that (En)∗X ∼ = (En)∗En as commutative (En) ∗ algebras had the homotopy type of BG, where G was an appropriate variant of the Morava stabilizer group. This is but one point of view on these results, and the reader should also consult [3], [38], and [41], among others. A point worth reiterating is that the moduli problems here begin with algebra: we have a homology theory E ∗ and a commutative ring A in E∗E comodules and we wish to discuss the homotopy type of the space T M(A) of all E∞ring spectra so that E∗X ∼ = A. We do not, a priori, assume that T M(A) is nonempty, or even that there is a spectrum X so that E∗X ∼ = A as comodules.
Realizability of algebraic Galois extensions by strictly commutative ring spectra
"... Abstract. We discuss some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are ..."
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Cited by 9 (6 self)
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Abstract. We discuss some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and GoerssHopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative Salgebras. Examples such as the complex Ktheory spectrum as a KOalgebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.
Γcohomology of rings of numerical polynomials and E∞ structures on
"... Abstract. We investigate Γcohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γcohomology vanishes above degree 1. As these cohomology groups are the obstruction group ..."
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Cited by 7 (6 self)
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Abstract. We investigate Γcohomology of some commutative cooperation algebras E∗E associated with certain periodic cohomology theories. For KU and E(1), the Adams summand at a prime p, and for KO we show that Γcohomology vanishes above degree 1. As these cohomology groups are the obstruction groups in the obstruction theory developed by Alan Robinson we deduce that these spectra admit unique E ∞ structures. As a consequence we obtain an E∞ structure for the connective Adams summand. For the JohnsonWilson spectrum E(n) with n � 1 we establish the existence of a unique E ∞ structure for its Inadic completion.
Co)homology theories for commutative Salgebras
"... The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually ..."
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Cited by 6 (3 self)
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The aim of this paper is to give an overview of some of the existing homology theories for commutative (S)algebras. We do not claim any originality; nor do we pretend to give a complete account. But the results in that field are widely spread in the literature, so for someone who does not actually work in that subject, it can be difficult to trace all the relationships between the different homology theories. The theories we aim to compare are • topological AndréQuillen homology • Gamma homology • stable homotopy of Γmodules • stable homotopy of algebraic theories • the AndréQuillen cohomology groups which arise as obstruction groups in the GoerssHopkins approach As a comparison between stable homotopy of Γmodules and stable homotopy of algebraic theories is not explicitly given in the literature, we will give a proof of Theorem 2.1 which says that both homotopy theories are isomorphic
Cohomology theories for highly structured ring spectra. arXiv: math.AT/0211275
"... Abstract. This is a survey paper on cohomology theories for A ∞ and E ∞ ring spectra. Different constructions and main properties of topological AndréQuillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to ..."
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Cited by 3 (1 self)
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Abstract. This is a survey paper on cohomology theories for A ∞ and E ∞ ring spectra. Different constructions and main properties of topological AndréQuillen cohomology and of topological derivations are described. We give sample calculations of these cohomology theories and outline applications to the existence of A ∞ and E ∞ structures on various spectra. We also explain the relationship between topological derivations, spaces of multiplicative maps and moduli spaces of multiplicative structures. 1.
REALISIBILITY OF ALGEBRAIC GALOIS EXTENSIONS BY STRICTLY COMMUTATIVE RING SPECTRA
, 2004
"... Abstract. We describe some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality a ..."
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Cited by 2 (1 self)
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Abstract. We describe some of the basic ideas of Galois theory for commutative Salgebras originally formulated by John Rognes. We restrict attention to the case of finite Galois groups. We describe the general framework developed by Rognes. Central rôles are played by the notion of strong duality and a trace or norm mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones applying obstruction theories of Robinson and GoerssHopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative Salgebras. Examples such as the Real Ktheory spectrum as a KOalgebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones and this leads to computable Harrison groups for such spectra. We consider the Tate spectrum associated to a GGalois extension and show that it is trivial, thus generalising an analogous result for algebraic Galois extensions. We end by proving an analogue of Hilbert’s theorem 90 for the units associated with a Galois extension.